Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #11677 |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C42 · Schur multiplier (H2) = C23 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C2×C4 | C2×#102 |
| commutator | C2×C8 | C2×C42 |
| Frattini, cosocle | C2×C4×C8 | C23 |
| socle | C22 | C2×#340 |
| prime radical | C24 | #103 |
| LCS | G ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C4 ◁ C22×C42 ◁ C22×C4×C8 ◁ G | |
| LECS | G ▷ C2×C4×C8 ▷ C2×C4 ▷ C2 ▷ C1 | |
481 conjugacy classes of subgroups.
| minimal degree | μ(G) = 24 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 3 generators on 24 points:
(1,2,4,8,5,7,3,6)(10,14)(11,18)(12,19)(15,17)(16,23)(20,24)(21,22);(2,6,7,8)(9,10,11,15,19,21,24,22,13,17,20,14,12,16,18,23);(2,7)(6,8)Polycyclic presentation:
2^9|a^2=e,b^2=f,d^2=gi,e^2=h,f^2=ghi,g^2=i,b^a=bd,c^a=ch,d^a=dg,f^a=fg,g^a=gi,d^b=di104 conjugacy classes — view the full character table.
| maximal subgroup of | #72321454, #72321464, #72321504 |
| maximal normal subgroup of | #72321454, #72321464, #72321504 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 34060);